To determine which expressions are equivalent to [tex]\( 25^x \)[/tex], let’s simplify and analyze each option step-by-step.
First, let’s rewrite [tex]\( 25^x \)[/tex] in a simpler form using exponents:
[tex]\[ 25^x \][/tex]
Since [tex]\( 25 \)[/tex] can be expressed as [tex]\( 5^2 \)[/tex]:
[tex]\[ 25^x = (5^2)^x \][/tex]
Using the properties of exponents, [tex]\((a^b)^c = a^{b \cdot c}\)[/tex]:
[tex]\[ (5^2)^x = 5^{2x} \][/tex]
So, [tex]\( 25^x \)[/tex] can be rewritten as:
[tex]\[ 5^{2x} \][/tex]
Now, let’s compare this form with each given option:
A. [tex]\( 5^2 \cdot 5^x \)[/tex]
- Using the properties of exponents, [tex]\( a^b \cdot a^c = a^{b+c} \)[/tex]:
[tex]\[ 5^2 \cdot 5^x = 5^{2+x} \][/tex]
- This is not equivalent to [tex]\( 25^x \)[/tex] or [tex]\( 5^{2x} \)[/tex].
B. [tex]\( (5 \cdot 5)^x \)[/tex]
- Simplify the expression inside the parentheses:
[tex]\[ 5 \cdot 5 = 25 \][/tex]
- So,
[tex]\[ (5 \cdot 5)^x = 25^x \][/tex]
- This is exactly the original expression, [tex]\( 25^x \)[/tex].
C. [tex]\( 5 \cdot 5^{2x} \)[/tex]
- Using the properties of exponents, [tex]\( a \cdot a^b = a^{1+b} \)[/tex]:
[tex]\[ 5 \cdot 5^{2x} = 5^{1+2x} \][/tex]
- This is not equivalent to [tex]\( 25^x \)[/tex] or [tex]\( 5^{2x} \)[/tex].
D. [tex]\( 5^x \cdot 5^x \)[/tex]
- Using the properties of exponents, [tex]\( a^b \cdot a^c = a^{b+c} \)[/tex]:
[tex]\[ 5^x \cdot 5^x = 5^{x+x} = 5^{2x} \][/tex]
- This is equivalent to [tex]\( 25^x \)[/tex] or [tex]\( 5^{2x} \)[/tex].
E. [tex]\( 5^{2x} \)[/tex]
- This is exactly the simplified form of [tex]\( 25^x \)[/tex]:
[tex]\[ 25^x = 5^{2x} \][/tex]
F. [tex]\( 5 \cdot 5^x \)[/tex]
- Using the properties of exponents, [tex]\( a \cdot a^b = a^{1+b} \)[/tex]:
[tex]\[ 5 \cdot 5^x = 5^{1+x} \][/tex]
- This is not equivalent to [tex]\( 25^x \)[/tex] or [tex]\( 5^{2x} \)[/tex].
Therefore, the expressions that are equivalent to [tex]\( 25^x \)[/tex] are:
- B. [tex]\( (5 \cdot 5)^x \)[/tex]
- D. [tex]\( 5^x \cdot 5^x \)[/tex]
- E. [tex]\( 5^{2x} \)[/tex]
